Approximate Lie symmetries and singular perturbation theory

Abstract

Perturbation theory plays a central role in the approximate solution of nonlinear differential equations. The resultant series expansions are usually divergent and require treatment by singular perturbation methods to generate uniformly valid solutions. However, applying these methods is a subtle art owing to the lack of globally applicable algorithms. Inspired by the fact that all exact solutions of differential equations are consequences of (Lie) symmetries, we reformulate perturbation theory for differential equations in terms of approximate symmetries, via expansions of the Lie symmetries of the solutions. This is a change in perspective from the usual method for obtaining series expansions of the solutions themselves. We show that these approximate symmetries are straightforward to calculate and are never singular; their integration is therefore a powerful way of constructing uniformly convergent solutions. This geometric viewpoint naturally implies that several key singular perturbation methods such as the general perturbative RG-inspired approach of Chen, Goldenfeld and Oono (CGO RG), the method of multiple scales (MMS), and the Poincare-Lindstedt method (PLM), exploit a fundamental class of approximate symmetries that we term ``hidden scale symmetries''. In turn, this clarifies when and why these methods succeed and just as importantly, when they fail. Our algorithmic method directly identifies and integrates these hidden scale symmetries, making it often simpler to implement, and permitting solution of problems where other methods are impractical. Finally, we show how other kinds of approximate symmetry can be exploited to solve systems that do not possess integrable hidden scale symmetries.Comment: 23 pages, 7 figure